Topological engineering of chiral anomalies in Janus nanoribbons

Supporting Information for Topological engineering of chiral anomalies in Janus nanoribbons Vasil A. Saroka,∗,†,‡Victor A. Demin,¶and Michele Pizzochero∗,§ †Department of Physics, University of Rome Tor Vergata and INFN, Via della Ricerca Scientifica 1, 00133 Roma, Italy ‡Institute for Nuclear Problems, Belarusian State University, Bobruiskaya 11, 220006 Minsk, Belarus ¶Emanuel Institute of Biochemical Physics RAS, 4 Kosygin Street, 119334 Moscow, Russia §Department of Physics, University of Bath, Bath BA2 7AY, United Kingdom E-mail: vasil.sarok[email protected]; [email protected] Supporting Note 1: Top-down topological description of Janus ribbons The purpose of this note is to link our bottom-up topological engineering to the top-down approaches known in the literature, in particular to the “weak” topological description in terms of parametric structures, which visualize the winding number or equivalently Zinvariant protected by the chiral symmetry. In what follows, we refer to such structures as Ryu-Hatsugai loops.1A few comments shall be made before we proceed. Firstly, although there is a quite general description,2we focus on the basic geometries. Secondly, the generalized description in Ref. 2 deals with the Zak phase and, as such, must be attributed to the Z2invariant. It seems that for an isotropic honeycomb lattice with 2 ×2 Hamiltonian S1 the situation is somewhat like for the 1D Su-Schrieffer-Heeger model,3i.e. the two definitions coincide and it is easy to mix and misname the two invariants. The parametric loops, however, clearly point on the Zinvariant; therefore, we stick to this viewpoint. Figure S1 presents the four basic GNR geometries for the honeycomb lattice and the derivation of the two periodic gauges for the graphene Hamiltonian. These two gauges give two possible phase portraits for the vector field in the reciprocal k-space as will be shown later. The geometries (1) zigzag, (2) bearded and (3) armchair have been considered in the seminal work.1The fourth twig geometry has been reported only recently.4It is important to note that Fig. S1 presents the geometries for the ribbon that can be fully tiled by the base unit cells of the 2D honeycomb lattice. In other words, all of them admit perfect matchings, in effect, each base unit cell represents a matching, and therefore zero graph deficit η. In Figure S1, we also show the GNRs with mixed edges. Those are incommensurable with the honeycomb lattice unit cell. They feature η= 0. Figure S2 summarized results for the basic GNR edges and their zero-energy modes on the honeycomb lattice. One shall notice that the two gauges identified in Fig S1 lead to two configurations of the vector field. Each such vector field can be parametrized by kx and ky, hence they lead to four possible parametric loops defined in the 3D-parameter space (Re[f],Im[f],0). The loops and their corresponding GNRs with the bands structures are depicted below in Fig. S2. It can be inferred from the band structures that zero-energy modes in zigzag and bearded geometries form a complementary pair that covers the full range of the Brillouin zone. The only difficulty is to resolve what happens at the Dirac point k= 2π/3, because in the wide-ribbon limit zero-energy modes may overlap at this point. To overcome this, we use the same approach as in the main text of the paper and introduce a small anisotropy to the lattice: t1= 1.1t2=t3. As can be seen in Figure S2, this anisotropy shifts k= 2π/3 loops for zigzag and bearded GNRs in opposite directions with respect to the origin of the parametric space. Thus, the two zero-energy modes of the ribbons indeed complement each other. A similar picture is observed for the GNRs with S2 1 1-2 1 2 2 1' 1'-2' 1' 1'' 1''-2'' 1'' 2' 2' 2'' 2'' armchair twig zigzag (2) (2) (1) (1) (3) (4) (1) (4) to (2) (3) to bearded half-bearded Janus GNR (1/2) (3/4) half-twig Janus GNR Figure S1. The GNR edge types (bold integer numbers) on a honeycomb lattice (thick gray) and the lattice tight-binding Hamiltonian Hin the two possible periodic gauges. a1and a2 are the primitive translations of the honeycomb lattice. t1,2,3are the hopping integrals for the three nearest neighbors. The base unit cells of the 2D lattice and their tillings resulting in the GNR unit cells are highlighted in light blue and dotted ovals. The neighboring unit cells of the 2D lattice contributing into the phase factor function fare highlighted with light green and light orange. The two atoms in the base and neighboring unit cells are enumerated with numbers and their primed versions, respectively. Janus ribbons unit cells each combining two basic edge geometries are shown by dashed boxes and labeled with bold fractions. S3 armchair and twig edges. The loop k= 0 for the armchair GNR transforms from a line into an ellipse encompassing the origin of the parametric space (cf. with Fig. 3(c) in Ref. 1), while similar loop for the twig edge ribbon shifts away from the origin. The complementary pairs of edges can be combined in the real space in a single Janus ribbon that forms either half-bearded GNR or a half-twig GNR, as can be seen in Fig. S1. For such ribbons, the periodic gauge for the parametric loops cannot be smoothly defined as a function of parameter kx,y to describe zero-energy states. The only possibility to define loops through the whole Brillouin zone is to divide this zone into two regions corresponding the zero -energy modes of complementary edges and to use the two different gauges accordingly. This situation is somewhat reminiscent of the Chern number tracing the obstruction with respect to the smooth gauge of the wavefunction on a 2D torus of the Brillouin zone.5This is a signature that Janus ribbons are linked to the “strong” topology. It is also important to note that, by bottom-up engineering, all the ribbons in Fig. S2 are trivial. Neither of those exhibits a fully flat band. However, the combination of complementary edges in a single Janus GNR gives rise to a fully flat band and “strong” topology that is in direct equivalence with the real space in situ Kekul´e patterns described by graph-theoretic deficit ηas presented in the main text of the paper. References (1) Ryu, S.; Hatsugai, Y. Topological Origin of Zero-Energy Edge States in Particle-Hole Symmetric Systems. Phys. Rev. Lett. 2002,89, 077002. (2) Delplace, P.; Ullmo, D.; Montambaux, G. Zak phase and the existence of edge states in graphene. Phys. Rev. B 2011,84, 195452. (3) Cayssol, J.; Fuchs, J. N. Topological and geometrical aspects of band theory. J. Phys. Mater. 2021,4, 034007. S4 (a) (d) (b) (f) (c) (e) Figure S2. The top-down topological description of GNR edges with Ryu-Hatsugai loops. (a, b) The vector field portraits corresponding to the two periodic gauges of f, i.e. (1) and (2) in Fig. S1. (c, d, e, f) The winding loops in parametric space for k= 0 (solid), 2π/3 (dashed), −2π/3 (dot-dashed) [up to 2/√3 factor in armchair and twig cases]. k≡kx for zigzag and bearded GNRs, while k≡kyfor twig and armchair GNRs. In each case, the loop parameter is a complementary one from (kx, ky) pair, which is changing along the lines of the corresponding style in (a) and (b). The band structures and their corresponding ribbons are shown to the left of each loop graphic. See Supplementary Data for details of calculations []. S5 (4) Xia, S.; Liang, Y.; Tang, L.; Song, D.; Xu, J.; Chen, Z. Photonic realization of a generic type of graphene edge states exhibiting topological flat band. Phys. Rev. Lett. 2023, 131, 013804. (5) Benevig, B. A.; Hughes, T. L. Topological Insulators and Topological Superconductors, 2nd ed.; Princeton University Press: Princeton, New Jersey, 2013. S6